Square and Square Roots

Squares

Introduction to Square Numbers

If a natural number m can be expressed as n², where n is also a natural number, then m is a square number.
Example: 1, 4, 9, 16 and 25.

Finding the Square of a Number

If n is a number, then its square is given as n×n=n². For example: Square of 5 is equal to 5×5=25

Properties of Square Numbers

Properties of square numbers are:

• If a number has 0, 1, 4, 5, 6 or 9 in the unit’s place, then it may or may not be a square number. If a number has 2, 3, 7 or 8 in its units place then it is not a square number.
• If a number has 1 or 9 in unit’s place, then it’s square ends in 1.
• If a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit’s place.
• For example, consider the number 64. The unit digit of 64 is 4 and it is a square number. Because the square of 8 is 64 and 64 is considered to be a square number.
• Consider a number 11 (i.e., the unit’s place of 11 is 1). Thus, the square of 11 is 121. Hence, square number 121 also has 1 in the unit’s place.

Finding square of a number with unit’s place 5

The square of a number N5 is equal to (N(N+1))×100+25, where N can have one or more than one digits.
For example: If N = 1, then 15² = (1 × 2) × 100 + 25
For example: If N = 1, then 15² = 200 + 25 = 225
If N = 20, then 205² = (20 × 21) × 100 + 25
If N = 20, then 205² = 42000 + 25 = 42025 For example: If N = 1, then
15² = (1 × 2) × 100 + 25
15² = 200 + 25 = 225
If N = 20, then
205² = (20 × 21) × 100 + 25
205² = 42000 + 25 = 42025

Perfect Squares

A number which is obtained from square of the other number is called perfect squares. For example, 81 is a perfect square number, which is obtained by taking the square of the number 9.

Interesting Patterns

There exists interesting patterns in:

• Adding triangular numbers
• Numbers between square numbers
• Adding odd numbers
• A sum of consecutive natural numbers
• Product of two consecutive even or odd natural numbers

Adding Triangular Numbers

Triangular numbers: It is a sequence of the numbers 1, 3, 6, 10, 15 etc. It is obtained by continued summation of the natural numbers. The dot pattern of a triangular number can be arranged as triangles.

If we add two consecutive triangular numbers, we get a square number. Example: 1 + 3 = 4 = 2² and 3 + 6 = 9 = 3²

Numbers between Square Numbers

There are 2n non-perfect square numbers between squares of the numbers n and (n + 1), where n is any natural number.

Example:

• There are two non-perfect square numbers (2, 3) between 1² = 1 and 2² = 4.
• There are four non-perfect square numbers (5, 6, 7, 8) between 2² = 4 and 3² = 9.

Addition of Odd Numbers

The sum of first n odd natural numbers is n².
Example:
1+3=4=2²
1+3+5=9=3²

Square of an odd number as a sum

Square of an odd number n can be expressed as sum of two consecutive positive integers n² - 12 and n² + 12
For example:
3² = 9 = 4 + 5 = 3² - 12 + 3² + 12
Similarly,
5² = 25 = 12 + 13 = 5² - 12 + 5² + 12

Square of an odd number as a sum

The product of two even or odd natural number can be calculated as, (a+1)×(a−1)=(a²−1), where a is a natural number, and a−1, a+1, are the consecutive odd or even numbers.
For example:
11 × 13 = (12 − 1) × (12 + 1)
11 × 13 = 12² − 1
11 × 13 = 144 − 1 = 143

Random Interesting Patterns Followed by Square Numbers

1² = 1
=11² = 121
= =111² = 12321
= = =1111² = 1234321
= = = =11111² = 123454321
= = = = =111111² = 12345654321
Patterns in numbers like 6, 67, 667, … :
7² = 49
=67² = 4489
==667² = 444889
===6667² = 44448889
====66667² = 4444488889
=====666667² = 444444888889

Pythagorean Triplets

For any natural number m > 1, we have (2m)² + (m² − 1)² = (m² + 1)².
2m, (m² − 1) and (m² + 1) forms a Pythagorean triplet. For m = 2, 2m = 4, m² − 1 = 3 and m² + 1 = 5. So, 3, 4, 5 is the required Pythagorean triplet.

Square Roots

Finding Square Root by Long Division Method

Steps involved in finding the square root of 484 by Long division method:

Step 1: Place a bar over every pair of numbers starting from the digit at units place. If the number of digits in it is odd, then the left-most single-digit too will have a bar.

Step 2: Take the largest number as divisor whose square is less than or equal to the number on the extreme left. Divide and write quotient.

Step 3: Bring down the number which is under the next bar to the right side of the remainder.

Step 4: Double the value of the quotient and enter it with a blank on the right side.

Step 5: Guess the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.

The remainder is 0, therefore, √484 = 22.

Be More Curious!

Finding Pythagorean Triplets for Any Given Number

If we are given any member of a Pythagorean triplet, then we can find the Pythagorean triplet by using general form 2m, m² – 1, m² + 1.
For example, if we want to find the Pythagorean triplet whose smallest number is 8.
Let, m² − 1 = 8
Li⇒ m = 3
2m = 6 and m² + 1 = 10
The triplet is 6, 8, and 10.
But 8 is not the smallest number of this triplet.
So, we substitute 2m = 8
o w esubstitute⇒ m = 4
m² − 1 = 15 and m² + 1 = 17
Therefore, 8, 15, 17 is the required triplet.

Short Answer Questions(SAQs)

What are Perfect squares?

When an integer is multiplied (whole number, positive, negative or zero) times itself, the resulting product is called a square number or a perfect square.

What are some of the properties of Square numbers?

1. It can only end in 0,1,4,5,6,9
2. Square number of even numbers are even
3. Square number of odd numbers are odd

What does the Pythagorean theoem state?

The Pythagorean theorem states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle).

MCQs

1. What is the square of 13?

(a) 169

(b) 143

(c) 156

(d) 153

► (a) 169

2. √81 is equal to:

(a) 7

(b) 8

(c) 9

(d) 10

► (c) 9

3. Which of the following is a perfect square?

(a) 50

(b) 81

(c) 90

(d) 70

► (b) 81

4. The square root of 144 is:

(a) 10

(b) 11

(c) 12

(d) 13

► (c) 12

5. Square of 0 is:

(a) 0

(b) 1

(c) Undefined

(d) Not defined

► (a) 0

6. Which number is not a perfect square?

(a) 64

(b) 100

(c) 121

(d) 98

► (d) 98

7. How many digits can a perfect square end in?

(a) 2 or 3

(b) 0, 1, 4, 5, 6, or 9

(c) Any number

(d) 2, 4, 6, or 8

► (b) 0, 1, 4, 5, 6, or 9

8. Which of the following is NOT a perfect square?

(a) 121

(b) 144

(c) 169

(d) 150

► (d) 150

9. (−7)² is:

(a) −49

(b) 49

(c) −14

(d) 14

► (b) 49

10. √25 + √36 =

(a) 11

(b) 12

(c) 13

(d) 10

► (a) 11

11. What is the square root of 1?

(a) 0

(b) 1

(c) −1

(d) Both (b) and (c)

► (d) Both (b) and (c)

12. How many numbers between 1 and 100 are perfect squares?

(a) 10

(b) 9

(c) 8

(d) 11

► (a) 10

13. 17² =

(a) 289

(b) 278

(c) 288

(d) 294

► (a) 289

14. Which of the following numbers has a square root which is a whole number?

(a) 75

(b) 90

(c) 100

(d) 98

► (c) 100

15. Square root of 225 is:

(a) 14

(b) 15

(c) 16

(d) 17

► (b) 15

16. Which number is both a square and a cube?

(a) 16

(b) 36

(c) 64

(d) 49

► (c) 64

17. √0 =

(a) 0

(b) 1

(c) −1

(d) Not defined

► (a) 0

18. The square of an even number is:

(a) Always even

(b) Always odd

(c) Sometimes even

(d) Never even

► (a) Always even